Some Symmetry and Duality Theorems on Multiple Zeta(-Star) Values

In this paper, we provide a symmetric formula and a duality formula relating multiple zeta values and zeta-star values. We find that the summation ∑a+b=r−1(−1)aζ★(a+2,{2}p−1)ζ★({1}b+1,{2}q) equals ζ★({2}p,{1}r,{2}q)+(−1)r+1ζ★({2}q,r+2,{2}p−1). With the help of this equation and Zagier’s ζ★({2}p,3,{2}q) formula, we can easily determine ζ★({2}p,1,{2}q) and several interesting expressions.